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Theorem sqrtval 14451
Description: Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
Assertion
Ref Expression
sqrtval (𝐴 ∈ ℂ → (√‘𝐴) = (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem sqrtval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2783 . . . 4 (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
213anbi1d 1419 . . 3 (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
32riotabidv 6933 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ ((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) = (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
4 df-sqrt 14449 . 2 √ = (𝑦 ∈ ℂ ↦ (𝑥 ∈ ℂ ((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
5 riotaex 6935 . 2 (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) ∈ V
63, 4, 5fvmpt 6589 1 (𝐴 ∈ ℂ → (√‘𝐴) = (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1068   = wceq 1507  wcel 2050  wnel 3067   class class class wbr 4923  cfv 6182  crio 6930  (class class class)co 6970  cc 10327  0cc0 10329  ici 10331   · cmul 10334  cle 10469  2c2 11489  +crp 12198  cexp 13238  cre 14311  csqrt 14447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-iota 6146  df-fun 6184  df-fv 6190  df-riota 6931  df-sqrt 14449
This theorem is referenced by:  sqrt0  14456  resqrtcl  14468  resqrtthlem  14469  sqrtneg  14482  sqrtcl  14576  sqrtthlem  14577
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